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I have a problem reconciling the following two passages from the progression document:
Notice that a common preoccupation of high school mathematics, distinguishing functions from relations, is not in the Standards. Time normally spent on exercises involving the vertical line test, or searching lists of ordered pairs to find two with the same x-coordinate and different y-coordinate, can be reallocated elsewhere.
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The essential question when investigating functions is: “Does each element of the domain correspond to exactly one element in the range?”
I have no problem disposing with relations or with the vertical line test, but the first passage goes past that and suggests avoiding going over ordered pairs. The second passage does seem to suggest we want students to look for domain elements that are associated with more than one unique range element.
Does the restriction from the first passage apply to other representations of associations? Such representations may be a table of corresponding values, a graph, an equation, or a verbal description. The quantities in question can be abstract and they can be a part of a real world context. In every case we can ask whether any of the quantities is a function of the other. Are some of the cases I mentioned problematic in that manner? If so, could you please elaborate why?
A couple thoughts:
* First, remember that because something is not in the standards doesn’t mean teachers can’t address it. The main point about these topics not being in the standards is that the assessment folks should not be writing items that test whether students can apply the vertical line test or pick out sets of ordered pairs with a particular property. These are procedures that aren’t very interesting when extracted away from their reasoning purpose.
* However, students should be able to look at a graph of x=y^2 and note that, for example, the value x = 1 corresponds to y = 1 and also y = -1, so it is not the graph of a function. Note that this is not the same kind of argument as applying the vertical line test, because it connects back to the definition of a function. The argument, “The line x=1 intersects the graph in two places so the graph is not of a function” is a black-box explanation for most students–they are told that you do such-and-such, and you interpret the results in some way–it is like reading tea leaves or consulting the oracle, but does not constitute mathematical reasoning.
The problem with standard questions about functions that ask students to employ the vertical line test or to look at ordered pairs is that students almost never realize that these are fundamentally the same kind of investigation: if you took the list of ordered pairs and plotted them in the coordinate plane, applying the vertical line test amounts to the same thing as inspecting the ordered pairs and looking for x-values that correspond to different y-values. In other words, for most students, these are completely disconnected procedures rather than different manifestations of the same kind of mathematical argument, one that relies on the definition of a function to determine if a relation is a function. We want students to be able to reason from the definition of a function to determine if a relation is a function; we don’t care if they can enter the correct letter when prompted, “Apply the vertical line test and mark y or n for whether the graph shown is the graph of a function.”
Kristin has pretty much said it all. Here are a couple more thoughts. The vertical line test and tables of ordered pairs are tools in service of the understanding expressed in your second quotation above. As long as they remain tools, that’s fine (although personally I think the vertical line test is excessive codification of a simple visual observation). The first quotation you give refers to “time normally spent on exercises” on the vertical line test or tables of coordinate pairs. The key words here are “time” and “exercises.” Once a tool becomes the subject of a set of exercises devoted specifically to it, it becomes a topic in its own right, disconnected from the understanding it originally served (as Kristin says in her last paragraph).
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